Kutta, Tim; Dierickx, Gauthier; Dette, Holger Statistical inference for the slope parameter in functional linear regression. (English) Zbl 07633932 Electron. J. Stat. 16, No. 2, 5980-6042 (2022). Summary: In this paper, we consider the linear regression model \(Y=SX+\varepsilon\) with functional regressors and responses. We develop new inference tools to quantify deviations of the true slope \(S\) from a hypothesized operator \(S_0\) with respect to the Hilbert-Schmidt norm \(|||S-S_0|||^2\), as well as the prediction error \(\mathbb{E}\| SX-S_0 X\|^2\). Our analysis is applicable to functional time series and based on asymptotically pivotal statistics. This makes it particularly user-friendly, because it avoids the choice of tuning parameters inherent in long-run variance estimation or bootstrap of dependent data. We also discuss two sample problems as well as change point detection. Finite sample properties are investigated by means of a simulation study.Mathematically, our approach is based on a sequential version of the popular spectral cut-off estimator \(\hat{S}_N\) for \(S\). We prove that (sequential) plug-in estimators of the deviation measures are \(\sqrt{N}\)-consistent and satisfy weak invariance principles. These results rest on the smoothing effect of \(L^2\)-norms, that we exploit by a new proof-technique, the smoothness shift, which has potential applications in other fields. Cited in 1 Document MSC: 62R10 Functional data analysis 62M20 Inference from stochastic processes and prediction Keywords:functional linear regression; prediction error; relevant tests; spectral cut-off Software:fda (R) × Cite Format Result Cite Review PDF Full Text: DOI arXiv Link References: [1] Andersson, J. and J. Lillestøl (2010). Modeling and forecasting electricity consumption by functional data analysis. Journal of Energy Markets 3(1), 3-15. [2] Aston, J. A. and C. Kirch (2012). Detecting and estimating changes in dependent functional data. J. Multivariate Anal. 109, 204-220. · Zbl 1241.62121 [3] Aue, A., G. Rice, and O. Sönmez (2020). Structural break analysis for spectrum and trace of covariance operators. Environmetrics 31(1), e2617. e2617 env.2617. · Zbl 1545.62700 [4] Babii, A. (2020). Honest confidence sets in nonparametric IV regression and other ill-posed models. Econometric Theory 36(4), 658-706. · Zbl 1447.62042 [5] Benatia, D., M. Carrasco, and J.-P. Florens (2017). Functional linear regression with functional response. J. Econometrics 201(2), 269-291. · Zbl 1377.62111 [6] Berg, P., J. Haerter, P. Thejll, C. Piani, S. Hagemann, and J. Christensen (2009, 09). Seasonal characteristics of relationship between daily precipitation intensity and surface temperature. Journal of Geophysical Research 114. [7] Berkes, I., R. Gabrys, L. Horváth, and P. Kokoszka (2009). Detecting changes in the mean of functional observations. J. R. Stat. Soc. Ser. B. Stat. Methodol. 71(5), 927-946. · Zbl 1411.62153 [8] Berkes, I., L. Horváth, and G. Rice (2013). Weak invariance principles for sums of dependent random functions. Stochastic Process. Appl. 123(2), 385-403. · Zbl 1269.60040 [9] Berkson, J. (1938). Some difficulties of interpretation encountered in the application of the chi-square test. J. Amer. Statist. Assoc. 33(203), 526-536. · Zbl 0019.17701 [10] Bissantz, N. and H. Holzmann (2008). Statistical inference for inverse problems. Inverse Problems 24(3), 034009. · Zbl 1137.62325 [11] Bonner, S., N. Newlands, and N. Heckman (2014). Modeling regional impacts of climate teleconnections using functional data analysis. Environmental and Ecological Statistics 21, 1-26. [12] Bücher, A. and I. Kojadinovic (2013). A dependent multiplier bootstrap for the sequential empirical copula process under strong mixing. Bernoulli 22(2), 927-968. · Zbl 1388.62123 [13] Cardot, H., F. Ferraty, A. Mas, and P. Sarda (2003). Testing hypotheses in the functional linear model. Scand. J. Stat. 30(1), 241-255. · Zbl 1034.62037 [14] Cardot, H., F. Ferraty, and P. Sarda (2003). Spline estimators for the functional linear model. Statist. Sinica 13(3), 571-591. · Zbl 1050.62041 [15] Cardot, H., A. Goia, and P. Sarda (2004). Testing for no effect in functional linear regression models, some computational approaches. Comm. Statist. Simulation Comput. 33, 179-199. · Zbl 1058.62037 [16] Cardot, H., A. Mas, and P. Sarda (2007). CLT in functional linear regression models. Probab. Theory Related Fields 138, 325-361. · Zbl 1113.60025 [17] Cavalier, L. (2008). Nonparametric statistical inverse problems. Inverse Problems 24(3), 034004. · Zbl 1137.62323 [18] Constantinou, P., P. Kokoszka, and M. Reimherr (2017). Testing separability of space-time functional processes. Biometrika 104(2), 425-437. · Zbl 1506.62541 [19] Crambes, C. and A. Mas (2013). Asymptotics of prediction in functional linear regression with functional outputs. Bernoulli 19(5B), 2627-2651. · Zbl 1280.62084 [20] Dehling, H. (1983). Limit theorems for sums of weakly dependent banach space valued random variables. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 63(3), 393—432. · Zbl 0496.60004 [21] Dehling, H., T. Mikosch, and M. Sørensen (2002). Empirical process techniques for dependent data. Birkhäuser. · Zbl 1005.00016 [22] Dette, H., G. Dierickx, and T. Kutta (2021). Quantifying deviations from separability in space-time functional processes. · Zbl 07594083 [23] Dette, H., K. Kokot, and S. Volgushev (2020). Testing relevant hypotheses in functional time series via self-normalization. J. R. Stat. Soc. Ser. B. Stat. Methodol. 82(3), 629-660. · Zbl 07554768 [24] Dette, H. and T. Kutta (2021). Detecting structural breaks in eigensystems of functional time series. Electron. J. Stat. 15(1), 944-983. · Zbl 1471.62558 [25] Dudley, R. M. (1999). Uniform Central Limit Theorems. Cambridge University Press. · Zbl 0951.60033 [26] Dunford, N. and J. T. Schwartz (1958). Linear operators. Part I: General theory. New York, Interscience Publishers. · Zbl 0084.10402 [27] Engl, H. W., M. Hanke, and A. Neubauer (1996). Regularization of inverse problems. Dordrecht: Kluwer Academic Publishers Group. · Zbl 0859.65054 [28] Hall, P. and J. L. Horowitz (2007). Methodology and convergence rates for functional linear regression. Ann. Statist. 35(1), 70-91. · Zbl 1114.62048 [29] Hariz, S., J. Wylie, and Q. Zhang (2007). Optimal rate of convergence for nonparametric change-point estimators for nonstationary sequences. Ann. Statist. 35, 1802-1826. · Zbl 1147.62043 [30] Hilgert, N., A. Mas, and N. Verzelen (2013). Minimax adaptive tests for the functional linear model. Ann. Statist. 41(2), 838-869. · Zbl 1267.62059 [31] Hörmann, S. and L. Kidzinski (2012). A note on estimation in Hilbertian linear models. Scand. J. Stat. 42(1), 43-62. · Zbl 1364.62175 [32] Horváth, L., M. Hu˘sková, and P. Kokoszka (2010). Testing the stability of the functional autoregressive process. J. Multivariate Anal. 101(2), 353-367. · Zbl 1178.62099 [33] Horváth, L. and P. Kokoszka (2012). Inference for Functional Data with Applications. New York: Springer Series in Statistics. · Zbl 1279.62017 [34] Horváth, L., P. Kokoszka, and R. Reeder (2011). Estimation of the mean of functional time series and a two sample problem. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75, 103-122. · Zbl 07555440 [35] Horváth, L. and R. Reeder (2011). Detecting changes in functional linear models. J. Multivariate Anal. 111, 310-334. · Zbl 1259.62046 [36] Imaizumi, M. and K. Kato (2018). PCA-based estimation for functional linear regression with functional responses. J. Multivariate Anal. 163, 15-36. · Zbl 1499.62137 [37] Imaizumi, M. and K. Kato (2019). A simple method to construct confidence bands in functional linear regression. Statist. Sinica 29(4), 2055-2081. · Zbl 1434.62150 [38] James, G., J. Wang, and J. Zhu (2009, 08). Functional linear regression that’s interpretable. Annals of Statistics 37. · Zbl 1171.62041 [39] Jarušková, D. (2013). Testing for a change in covariance operator. J. Statist. Plann. Inference 143(9), 1500-1511. · Zbl 1279.62124 [40] Kokoszka, P. (2012). Dependent functional data. Int Sch Res Notices Probability and Statistics 2012. · Zbl 06169714 [41] Kokoszka, P. and M. Reimherr (2013). Asymptotic normality of the principal components of functional time series. Stochastic Process. Appl. 123(5), 1546-1562. · Zbl 1275.62066 [42] Kong, D., A.-M. Staicu, and A. Maity (2016). Classical testing in functional linear models. J. Nonparametr. Stat. 28(4), 813-838. PMID: 28955155. · Zbl 1348.62136 [43] Kuelbs, J. and W. Philip (1980). Almost sure invariance principles for partial sums of mixing b-valued random variables. The Annals of Probability 8(6), 1003-1036. · Zbl 0451.60008 [44] Liebl, D. (2013). Modeling and forecasting electricity spot prices: A functional data perspective. Ann. Appl. Stat. 7(3), 1562-1592. · Zbl 1454.62267 [45] Merlevède, F., M. Peligrad, and S. Utev (2006). Recent advances in invariance principles for stationary sequences. Probab. Surv. 3, 1-36. · Zbl 1189.60078 [46] Moricz, F. A., R. J. Serfling, and W. F. Stout (1982). Moment and probability bounds with quasi-superadditive structure for the maximum partial sum. Ann. Probab. 10(4), 1032-1040. · Zbl 0499.60052 [47] Politis, D. and J. Romano (1994). The stationary bootstrap. J. Amer. Statist. Assoc. 89, 1303-1313. · Zbl 0814.62023 [48] Qiao, X., S. Guo, and G. M. James (2019). Functional graphical models. J. Amer. Statist. Assoc. 114, 211-222. · Zbl 1478.62123 [49] Quesada, B., R. Vautard, P. Yiou, M. Hirschi, and S. Seneviratne (2012, 10). Asymmetric european summer heat predictability from wet and dry winters and springs. Nature Clim. Change 2, 736-741. [50] Ramsay, J. O. and B. W. Silverman (1997). Functional Data Analysis. Berlin, Springer. · Zbl 0882.62002 [51] Samur, J. D. (1984). Convergence of sums of mixing triangular arrays of random vectors with stationary rows. Ann. Probab. 12(2), 390-426. · Zbl 0542.60012 [52] Samur, J. D. (1987). On the invariance principle for stationary \(ϕ\)-mixing triangular arrays with infinitely divisible limits. Probab. Theory Related Fields 75(2), 245-259. · Zbl 0594.60039 [53] Scheipl, F. and S. Greven (2016). Identifiability in penalized function-on-function regression models. Electron. J. Stat. 10(1), 495-526. · Zbl 1332.62249 [54] Shao, X. (2015). Self-normalization for time series: A review of recent developments. J. Amer. Statist. Assoc. 110, 1797-1817. · Zbl 1373.62456 [55] Shin, H. and S. Lee (2016). An rkhs approach to robust functional linear regression. Statist. Sinica 26, 255-272. · Zbl 1372.62018 [56] Sørensen, H., J. Goldsmith, and L. Sangalli (2013). An introduction with medical applications to functional data analysis. Stat. Med. 32, 5222-5240. [57] Stöhr, C., J. Aston, and C. Kirch (2021). Detecting changes in the covariance structure of functional time series with application to fmri data. Econom. Stat., 44-62. [58] Trenberth, K. E., A. Dai, R. M. Rasmussen, and D. B. Parsons (2003). The changing character of precipitation. Bulletin of the American Meteorological Society 84(9), 1205-1218. [59] Trenberth, K. E. and D. J. Shea (2005). Relationships between precipitation and surface temperature. Geophysical Research Letters 32(14). [60] van der Vaart, A. W. and J. A. Wellner (1996). Weak convergence and empirical processes. With applications to statistics. New York: Springer Series in Statistics. · Zbl 0862.60002 [61] Weidmann, J. (1980). Linear Operators in Hilbert Spaces, Volume 68 of Graduate Texts in Mathematics. Berlin, New York, Springer. · Zbl 0434.47001 [62] Yao, F., H.-G. Müller, and J.-L. Wang (2005). Functional linear regression analysis for longitudinal data. Ann. Statist. 33(6), 2873-2903. · Zbl 1084.62096 [63] Yuan, M. and T. Cai (2012). A reproducing kernel hilbert space approach to functional linear regression. Ann. Statist. 38(6), 3412-3444. · Zbl 1204.62074 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.